Theorem riotaprop 5901

Description: Properties of a restricted definite description operator. Todo (df-riota 5877 update): can some uses of riota2f 5899 be shortened with this? (Contributed by NM, 23-Nov-2013.)

Hypotheses

Ref	Expression
riotaprop.0	|- F/ x ps
riotaprop.1	|- B = ( iota_ x e. A ph )
riotaprop.2	|- ( x = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
riotaprop	|- ( E! x e. A ph -> ( B e. A /\ ps ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)   B( x)

Proof of Theorem riotaprop

Step	Hyp	Ref	Expression
1		riotaprop.1	. . 3 |- B = ( iota_ x e. A ph )
2		riotacl 5892	. . 3 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. A )
3	1, 2	eqeltrid 2283	. 2 |- ( E! x e. A ph -> B e. A )
4	1	eqcomi 2200	. . . 4 |- ( iota_ x e. A ph ) = B
5		nfriota1 5885	. . . . . 6 |- F/_ x ( iota_ x e. A ph )
6	1, 5	nfcxfr 2336	. . . . 5 |- F/_ x B
7		riotaprop.0	. . . . 5 |- F/ x ps
8		riotaprop.2	. . . . 5 |- ( x = B -> ( ph <-> ps ) )
9	6, 7, 8	riota2f 5899	. . . 4 |- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )
10	4, 9	mpbiri 168	. . 3 |- ( ( B e. A /\ E! x e. A ph ) -> ps )
11	3, 10	mpancom 422	. 2 |- ( E! x e. A ph -> ps )
12	3, 11	jca 306	1 |- ( E! x e. A ph -> ( B e. A /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   F/wnf 1474   e. wcel 2167   E!wreu 2477   iota_crio 5876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-uni 3840  df-iota 5219  df-riota 5877

This theorem is referenced by:  lble  8974